Embedded newform 69.3.f.a.19.4

• Label: 69.3.f.a.19.4
• Level: $69$
• Weight: $3$
• Character: 69.19
• Analytic conductor: $1.880$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	69.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.88011382409\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(8\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label			19.4
Character	\(\chi\)	\(=\)	69.19
Dual form			69.3.f.a.40.4

$q$-expansion

\(f(q)\)	\(=\)	\(q+(0.0585879 + 0.128290i) q^{2} +(-1.66189 + 0.487975i) q^{3} +(2.60642 - 3.00797i) q^{4} +(0.992973 - 1.54510i) q^{5} +(-0.159969 - 0.184614i) q^{6} +(8.01382 - 1.15221i) q^{7} +(1.07988 + 0.317082i) q^{8} +(2.52376 - 1.62192i) q^{9} +O(q^{10})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 80 q + 4 q^{2} - 12 q^{6} - 4 q^{8} - 24 q^{9}+O(q^{10}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/69\mathbb{Z}\right)^\times\).

\(n\)	\(28\)	\(47\)
\(\chi(n)\)	\(e\left(\frac{15}{22}\right)\)	\(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)			\(a_p / p^{(k-1)/2}\)			\( \alpha_p\)			\( \theta_p \)
\(2\)	0.0585879	+	0.128290i	0.0292939	+	0.0641448i	0.923713	−	0.383084i	\(-0.125138\pi\)
							−0.894419	+	0.447229i	\(0.852411\pi\)
\(3\)	−1.66189	+	0.487975i	−0.553964	+	0.162658i
							\(5\)	0.992973	−	1.54510i	0.198595	−	0.309019i	−0.727645	−	0.685954i	\(-0.759385\pi\)
0.926240	+	0.376934i	\(0.123022\pi\)
\(7\)	8.01382	−	1.15221i	1.14483	−	0.164602i	0.456312	−	0.889820i	\(-0.349170\pi\)
							0.688519	+	0.725218i	\(0.258261\pi\)
\(11\)	0.749080	+	0.342094i	0.0680982	+	0.0310994i	0.449173	−	0.893445i	\(-0.351719\pi\)
							−0.381075	+	0.924544i	\(0.624446\pi\)
\(13\)	−0.162100	+	1.12743i	−0.0124692	+	0.0867253i	−0.995106	−	0.0988156i	\(-0.968495\pi\)
							0.982637	+	0.185541i	\(0.0594037\pi\)
\(17\)	−17.4391	+	15.1110i	−1.02583	+	0.888884i	−0.993864	−	0.110605i	\(-0.964721\pi\)
							−0.0319621	+	0.999489i	\(0.510176\pi\)
\(19\)	−5.19015	−	4.49729i	−0.273166	−	0.236699i	0.507495	−	0.861655i	\(-0.330572\pi\)
							−0.780660	+	0.624955i	\(0.785117\pi\)
\(23\)	−22.1729	+	6.11252i	−0.964039	+	0.265762i
							\(29\)	19.4760	+	22.4765i	0.671587	+	0.775052i	0.984624	−	0.174689i	\(-0.0558919\pi\)
−0.313037	+	0.949741i	\(0.601346\pi\)
\(31\)	−13.9557	−	4.09775i	−0.450183	−	0.132186i	0.0487790	−	0.998810i	\(-0.484467\pi\)
							−0.498962	+	0.866624i	\(0.666285\pi\)
\(37\)	23.6533	+	36.8052i	0.639278	+	0.994735i	0.998117	+	0.0613361i	\(0.0195361\pi\)
							−0.358840	+	0.933399i	\(0.616827\pi\)
\(41\)	−32.4780	−	20.8724i	−0.792147	−	0.509082i	0.0808974	−	0.996722i	\(-0.474221\pi\)
							−0.873045	+	0.487640i	\(0.837858\pi\)
\(43\)	−5.48506	−	18.6804i	−0.127560	−	0.434428i	0.870803	−	0.491632i	\(-0.163600\pi\)
							−0.998362	+	0.0572043i	\(0.981781\pi\)
\(47\)	5.74820			0.122302			0.0611510	−	0.998129i	\(-0.480523\pi\)
							0.0611510	+	0.998129i	\(0.480523\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.3.f.a.19.4	✓	80
3.2	odd	2		207.3.j.b.19.5		80
23.17	odd	22	inner	69.3.f.a.40.4	yes	80
69.17	even	22		207.3.j.b.109.5		80

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.3.f.a.19.4	✓	80	1.1	even	1	trivial
69.3.f.a.40.4	yes	80	23.17	odd	22	inner
207.3.j.b.19.5		80	3.2	odd	2
207.3.j.b.109.5		80	69.17	even	22